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The transaction fee is processed by and received by the bitcoin miner. When a new bitcoin block is generated with a successful hash, the information for all of the transactions is included with the block and all transaction fees are collected by that user creating the block, who is free to assign those fees to himself.

So yes, the fee is collected by the miner of the discovered block. If you look to a random block on the blockchain site you see that the fees are 0.03334571 BTC (in that specific block).

The transaction fee you pay could be different depending on which wallet software you use. For example the fee from multibit is 0.0001 BTC. You can also pay no fees, but then there is a chance that your transaction takes a lot of blocks to be included in the blockchain.